The Banach spaces consist of covariant functions on locally compact groups associated to continuous characters (one-dimensional continuous irreducible unitary representations) of closed subgroups appear in variant mathematical areas and their applications including calculus of pseudo-differential operators, number theory (automorphic forms), induced representations,
homogenuos spaces, complex (hypercomplex) analysis, theoretical aspects of mathematical physics including coherent states and covariant analysis, see [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}.
